Celia Hoyles

Exploring mathematics through construction and collaboration

All learning environments are designed based on a set of epistemological assumptions about what knowledge should be learned. For example, most mathematics classrooms are designed to teach a certain kind of mathematical knowledge that comprises procedures that solve isolated problems quickly, and this implicitly devalues the importance of structural understanding or, put another way, of developing an appreciation of underlying mathematical models (see Lehrer & Schauble, this volume). This means that students all too often do not appreciate the need for consistency or rigor, so do not notice conflicts, and therefore cannot learn from it. Based on our research in a variety of workplace situations, we are convinced that a crucial element of knowledge required by most, if not all, people, is precisely this appreciation of underlying models. A version of mathematics that emphasizes structures has also the potential to help students understand the computational systems that are increasingly critical in todays society, because computer systems are mathematical models—computer software is built out of variables and relationships. As technology becomes more and more advanced, and the underlying models become more and more obscure and invisible, it becomes increasingly important that children learn awareness of models; how to build, revise and evaluate them, and to develop some analytic understanding of how inputs relate to outputs. In this chapter, we describe two learning environments that we have designed to further this agenda. Each of these environments is based on two principles. The first is constructionism: we should put learners in situations where they can construct and revise their own models (see Kafai, and Lehrer & Schauble, this volume). The second is collaboration: if our concern is that students come to understand what is significant about